Applications of finite continued fractions

In knot theory continued fractions are used to classify rational tangles. Conway proved that two rational tangles are isotopic if and only if they have the same fraction. This is proved by Kauffman in http://arxiv.org/pdf/math/0311499.pdf. The paper also contains all the basic definitions and I think it can be read by any mathematician.

You did not limit the context of continued fractions to numbers. Did you ? Then continued fractions can be used whenever you have a Euclidian division, preferably when there is a natural choice of quotient / remainder, so that it is done in a unique way. An important example is that of polynomials. Then continued fractions can be used to find accurate approximations of smooth functions by rational fractions about a given point, say $x=0$. This is related to Padé approximants.

This is described in the French wikipedia page (sorry, not in the English one) link text

One of the first factorization algorithms beyond trial division and Fermat's method was CFRAC: from the continued fraction expansion of $\sqrt{n}$ one computed solutions $x^2 - ny^2 = d^2$ and then had the (possibly trivial) factor $\gcd(n,x-d)$ of $n$. It is the father of the quadratic seive method.